Symmetries and Independence
in Noncommutative Probability
Claus Köstler
Institute of Mathematics and Physics
Aberystwyth University
[email protected]
Stochastic Analysis Seminar Series
Maths Institute and Oxford-Man Institute
University of Oxford
January 18, 2010
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivation
Though many probabilistic symmetries are conceivable [...], four of
them - stationarity, contractability, exchangeablity and
rotatability - stand out as especially interesting and important in
several ways: Their study leads to some deep structural theorems
of great beauty and significance [...].
Olav Kallenberg (2005)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivation
Though many probabilistic symmetries are conceivable [...], four of
them - stationarity, contractability, exchangeablity and
rotatability - stand out as especially interesting and important in
several ways: Their study leads to some deep structural theorems
of great beauty and significance [...].
Olav Kallenberg (2005)
Question:
Can one transfer the related concepts to noncommutative
probability and do they turn out to be fruitful in the study of the
structure of operator algebras and quantum dynamics?
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivation
Though many probabilistic symmetries are conceivable [...], four of
them - stationarity, contractability, exchangeablity and
rotatability - stand out as especially interesting and important in
several ways: Their study leads to some deep structural theorems
of great beauty and significance [...].
Olav Kallenberg (2005)
Question:
Can one transfer the related concepts to noncommutative
probability and do they turn out to be fruitful in the study of the
structure of operator algebras and quantum dynamics?
Remark
Noncommutative probability = classical & quantum probability
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Hierarchy of distributional symmetries
invariant objects
stationary
contractable
exchangeable
rotatable
Claus Köstler
transformations
shifts
sub-sequences
permutations
isometries
Symmetries and Independence in Noncommutative Probability
Hierarchy of distributional symmetries
invariant objects
stationary
contractable
exchangeable
rotatable
transformations
shifts
sub-sequences
permutations
isometries
Topic of this talk:
• invariant objects are generated by an
infinite sequence of random variables
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Hierarchy of distributional symmetries
invariant objects
stationary
contractable
exchangeable
rotatable
transformations
shifts
sub-sequences
permutations
isometries
Topic of this talk:
• invariant objects are generated by an
infinite sequence of random variables
• only the first three symmetries are considered
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Hierarchy of distributional symmetries
invariant objects
stationary
contractable
exchangeable
rotatable
transformations
shifts
sub-sequences
permutations
isometries
Topic of this talk:
• invariant objects are generated by an
infinite sequence of random variables
• only the first three symmetries are considered
• contractable = spreadable ( = subsymmetric)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivating example for de Finetti’s theorem
”Any exchangeable process is an average of i.i.d. processes.”
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivating example for de Finetti’s theorem
”Any exchangeable process is an average of i.i.d. processes.”
Theorem (De Finetti 1931)
Let X1 , X2 , . . . infinite sequence of {0, 1}-valued r.v.’s such that
P(X1 = e1 , . . . , Xn = en ) = P(Xπ(1) = e1 , . . . , Xπ(n) = en )
holds for all n ∈ N, permutations π and every e1 , . . . , en ∈ {0, 1}.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivating example for de Finetti’s theorem
”Any exchangeable process is an average of i.i.d. processes.”
Theorem (De Finetti 1931)
Let X1 , X2 , . . . infinite sequence of {0, 1}-valued r.v.’s such that
P(X1 = e1 , . . . , Xn = en ) = P(Xπ(1) = e1 , . . . , Xπ(n) = en )
holds for all n ∈ N, permutations π and every e1 , . . . , en ∈ {0, 1}.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Motivating example for de Finetti’s theorem
”Any exchangeable process is an average of i.i.d. processes.”
Theorem (De Finetti 1931)
Let X1 , X2 , . . . infinite sequence of {0, 1}-valued r.v.’s such that
P(X1 = e1 , . . . , Xn = en ) = P(Xπ(1) = e1 , . . . , Xπ(n) = en )
holds for all n ∈ N, permutations π and every e1 , . . . , en ∈ {0, 1}.
Then there exists a unique probability measure ν on [0, 1] such that
Z
P(X1 = e1 , . . . , Xn = en ) = p s (1 − p)n−s dν(p),
where s = e1 + e2 + . . . + en .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
- TODAY: Random variables X are elements of L∞ (A)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
- TODAY: Random variables X are elements of L∞ (A)
• A (noncommutative) probability space (A, ϕ) consists of
- a von Neumann algebra A acting on some Hilbert space H
- a faithful normal state ϕ on A
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
- TODAY: Random variables X are elements of L∞ (A)
• A (noncommutative) probability space (A, ϕ) consists of
- a von Neumann algebra A acting on some Hilbert space H
- a faithful normal state ϕ on A
- TODAY: Random variables are selfadjoint operators in A.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
- TODAY: Random variables X are elements of L∞ (A)
• A (noncommutative) probability space (A, ϕ) consists of
- a von Neumann algebra A acting on some Hilbert space H
- a faithful normal state ϕ on A
- TODAY: Random variables are selfadjoint operators in A.
• The automorphisms of a probability space Aut(A, ϕ) are
*-automorphisms α of A with ϕ ◦ α = ϕ.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Terminology of noncommutative probability
• Consider a standard probability space A := (Ω, Σ, µ) with the
R
expectation ϕ(X ) :=
Ω X (ω) dµ(ω).
∞
The pair
L (A), ϕ
is called a classical (or commutative) probability space.
- L∞ (A) is a von Neumann algebra acting on L2 (A)
- ϕ is a faithful normal state on L∞ (A)
- TODAY: Random variables X are elements of L∞ (A)
• A (noncommutative) probability space (A, ϕ) consists of
- a von Neumann algebra A acting on some Hilbert space H
- a faithful normal state ϕ on A
- TODAY: Random variables are selfadjoint operators in A.
• The automorphisms of a probability space Aut(A, ϕ) are
*-automorphisms α of A with ϕ ◦ α = ϕ.
• BEYOND TODAY: a more general approach uses certain
*-homomorphisms as random variables (Idea: ιX (f ) = f ◦ X )
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributions
Given the probability space (A, ϕ),
two sequences (xn )n≥0 and (yn )n≥0 in A have the same
distribution if
ϕ xi(1) xi(2) · · · xi(n) = ϕ yi(1) yi(2) · · · yi(n)
for all n-tuples i : {1, 2, . . . , n} → N0 and n ∈ N.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributions
Given the probability space (A, ϕ),
two sequences (xn )n≥0 and (yn )n≥0 in A have the same
distribution if
ϕ xi(1) xi(2) · · · xi(n) = ϕ yi(1) yi(2) · · · yi(n)
for all n-tuples i : {1, 2, . . . , n} → N0 and n ∈ N.
Notation
distr
(x0 , x1 , x2 , . . .) = (y0 , y1 , y2 , . . .)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributional symmetries
Just as in the classical case we can now talk about distributional
symmetries. A sequence (xn )n≥0 is
distr
• exchangeable if (x0 , x1 , x2 , . . .) = (xπ(0) , xπ(1) , xπ(2) , . . .) for
any (finite) permutation π ∈ S∞ of N0 .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributional symmetries
Just as in the classical case we can now talk about distributional
symmetries. A sequence (xn )n≥0 is
distr
• exchangeable if (x0 , x1 , x2 , . . .) = (xπ(0) , xπ(1) , xπ(2) , . . .) for
any (finite) permutation π ∈ S∞ of N0 .
distr
• spreadable if (x0 , x1 , x2 , . . .) = (xn0 , xn1 , xn2 , . . .) for any
subsequence (n0 , n1 , n2 , . . .) of (0, 1, 2, . . .).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributional symmetries
Just as in the classical case we can now talk about distributional
symmetries. A sequence (xn )n≥0 is
distr
• exchangeable if (x0 , x1 , x2 , . . .) = (xπ(0) , xπ(1) , xπ(2) , . . .) for
any (finite) permutation π ∈ S∞ of N0 .
distr
• spreadable if (x0 , x1 , x2 , . . .) = (xn0 , xn1 , xn2 , . . .) for any
subsequence (n0 , n1 , n2 , . . .) of (0, 1, 2, . . .).
distr
• stationary if (x0 , x1 , x2 , . . .) = (xk , xk+1 , xk+2 , . . .) for all
k ∈ N.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributional symmetries
Just as in the classical case we can now talk about distributional
symmetries. A sequence (xn )n≥0 is
distr
• exchangeable if (x0 , x1 , x2 , . . .) = (xπ(0) , xπ(1) , xπ(2) , . . .) for
any (finite) permutation π ∈ S∞ of N0 .
distr
• spreadable if (x0 , x1 , x2 , . . .) = (xn0 , xn1 , xn2 , . . .) for any
subsequence (n0 , n1 , n2 , . . .) of (0, 1, 2, . . .).
distr
• stationary if (x0 , x1 , x2 , . . .) = (xk , xk+1 , xk+2 , . . .) for all
k ∈ N.
distr
• identically distributed if (xk , xk , xk , . . .) = (xl , xl , xl , . . .)
for all k, l ∈ N0
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative distributional symmetries
Just as in the classical case we can now talk about distributional
symmetries. A sequence (xn )n≥0 is
distr
• exchangeable if (x0 , x1 , x2 , . . .) = (xπ(0) , xπ(1) , xπ(2) , . . .) for
any (finite) permutation π ∈ S∞ of N0 .
distr
• spreadable if (x0 , x1 , x2 , . . .) = (xn0 , xn1 , xn2 , . . .) for any
subsequence (n0 , n1 , n2 , . . .) of (0, 1, 2, . . .).
distr
• stationary if (x0 , x1 , x2 , . . .) = (xk , xk+1 , xk+2 , . . .) for all
k ∈ N.
distr
• identically distributed if (xk , xk , xk , . . .) = (xl , xl , xl , . . .)
for all k, l ∈ N0
Lemma (Hierarchy of distributional symmetries)
Exchangeability ⇒ Spreadability ⇒ Stationarity ⇒ Identical distr.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative conditional independence
Let A0 , A1 , A2 be three von Neumann subalgebras of A with
ϕ-preserving conditional expectations Ei : A → Ai (i = 0, 1, 2).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative conditional independence
Let A0 , A1 , A2 be three von Neumann subalgebras of A with
ϕ-preserving conditional expectations Ei : A → Ai (i = 0, 1, 2).
Definition
A1 and A2 are A0 -independent if
E0 (xy ) = E0 (x)E0 (y )
Claus Köstler
(x ∈ A0 ∨ A1 , y ∈ A0 ∨ A2 )
Symmetries and Independence in Noncommutative Probability
Noncommutative conditional independence
Let A0 , A1 , A2 be three von Neumann subalgebras of A with
ϕ-preserving conditional expectations Ei : A → Ai (i = 0, 1, 2).
Definition
A1 and A2 are A0 -independent if
E0 (xy ) = E0 (x)E0 (y )
(x ∈ A0 ∨ A1 , y ∈ A0 ∨ A2 )
Remarks
• If A = L∞ (A) we obtain conditional independence with
respect to a sub-σ-algebra.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative conditional independence
Let A0 , A1 , A2 be three von Neumann subalgebras of A with
ϕ-preserving conditional expectations Ei : A → Ai (i = 0, 1, 2).
Definition
A1 and A2 are A0 -independent if
E0 (xy ) = E0 (x)E0 (y )
(x ∈ A0 ∨ A1 , y ∈ A0 ∨ A2 )
Remarks
• If A = L∞ (A) we obtain conditional independence with
respect to a sub-σ-algebra.
• Many different forms of noncommutative independence!
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative conditional independence
Let A0 , A1 , A2 be three von Neumann subalgebras of A with
ϕ-preserving conditional expectations Ei : A → Ai (i = 0, 1, 2).
Definition
A1 and A2 are A0 -independent if
E0 (xy ) = E0 (x)E0 (y )
(x ∈ A0 ∨ A1 , y ∈ A0 ∨ A2 )
Remarks
• If A = L∞ (A) we obtain conditional independence with
respect to a sub-σ-algebra.
• Many different forms of noncommutative independence!
• C-independence & Speicher’s universality rules
tensor independence or free independence
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Conditional independence of sequences
A sequence (xn )n∈N0 in A is (full) B-independent if
_
_
{xi | i ∈ I } ∨ B and
{xj | j ∈ J} ∨ B
are B-independent whenever I ∩ J = ∅ with I , J ⊂ N0 .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Conditional independence of sequences
A sequence (xn )n∈N0 in A is (full) B-independent if
_
_
{xi | i ∈ I } ∨ B and
{xj | j ∈ J} ∨ B
are B-independent whenever I ∩ J = ∅ with I , J ⊂ N0 .
Remark
• B may not be contained in
Claus Köstler
W
{xi | i ∈ I }
Symmetries and Independence in Noncommutative Probability
Conditional independence of sequences
A sequence (xn )n∈N0 in A is (full) B-independent if
_
_
{xi | i ∈ I } ∨ B and
{xj | j ∈ J} ∨ B
are B-independent whenever I ∩ J = ∅ with I , J ⊂ N0 .
Remark
• B may not be contained in
W
{xi | i ∈ I }
W
• Example of mixed coin tosses: dim {xi } = 2, but
B ' L∞ ([0, 1], ν) may be infinite dimensional!
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Classical dual version of extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(c) (xn )n∈N0 is spreadable
(e) (xn )n∈N0 is Atail -independent and identically distributed.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Classical dual version of extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(c) (xn )n∈N0 is spreadable
(e) (xn )n∈N0 is Atail -independent and identically distributed.
Theorem (De Finetti ’31, Ryll-Nardzewski ’57,..., K. ’08)
A ' L∞ (A) implies:
(a) ⇔ (c) ⇔ (e)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Classical dual version of extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(c) (xn )n∈N0 is spreadable
(e) (xn )n∈N0 is Atail -independent and identically distributed.
Theorem (De Finetti ’31, Ryll-Nardzewski ’57,..., K. ’08)
A ' L∞ (A) implies:
(a) ⇔ (c) ⇔ (e)
Remark
Here tail algebra is commutative
Claus Köstler
ergodic decompositions
Symmetries and Independence in Noncommutative Probability
Noncommutative De Finetti theorems
with commutativity conditions
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(e) (xn )n∈N0 is identically distributed and Atail -independent
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative De Finetti theorems
with commutativity conditions
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(e) (xn )n∈N0 is identically distributed and Atail -independent
Theorem (Størmer ’69,... Hudson ’76, ... Petz ’89,... K ’08)
If the xn ’s mutually commute:
Claus Köstler
(a) ⇔ (e)
Symmetries and Independence in Noncommutative Probability
Noncommutative De Finetti theorems
with commutativity conditions
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
n≥0 k≥n
and consider:
(a) (xn )n∈N0 is exchangeable
(e) (xn )n∈N0 is identically distributed and Atail -independent
Theorem (Størmer ’69,... Hudson ’76, ... Petz ’89,... K ’08)
If the xn ’s mutually commute:
(a) ⇔ (e)
Theorem (Accardi & Lu ’93, ..., K. ’08)
If Atail is abelian:
(a) ⇒ (e)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
and
(a)
(c)
(d)
(e)
consider:
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
n≥0 k≥n
is
is
is
is
exchangeable
spreadable
stationary and Atail -independent
identically distributed and Atail -independent
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
and
(a)
(c)
(d)
(e)
consider:
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
n≥0 k≥n
is
is
is
is
exchangeable
spreadable
stationary and Atail -independent
identically distributed and Atail -independent
)
Theorem (K. ’07-’08
(a) ⇒ (c) ⇒ (d) ⇒ (e),
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
and
(a)
(c)
(d)
(e)
consider:
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
n≥0 k≥n
is
is
is
is
exchangeable
spreadable
stationary and Atail -independent
identically distributed and Atail -independent
Theorem (K. ’07-’08, Gohm & K. ’08)
(a) ⇒ (c) ⇒ (d) ⇒ (e),
but (a) 6⇐ (c) 6⇐ (d) 6⇐ (e)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Noncommutative extended De Finetti theorem
Let (xn )n∈N0 be random variables in (A, ϕ) with tail algebra
\ _
Atail :=
{xk },
and
(a)
(c)
(d)
(e)
consider:
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
(xn )n∈N0
n≥0 k≥n
is
is
is
is
exchangeable
spreadable
stationary and Atail -independent
identically distributed and Atail -independent
Theorem (K. ’07-’08, Gohm & K. ’08)
(a) ⇒ (c) ⇒ (d) ⇒ (e),
but (a) 6⇐ (c) 6⇐ (d) 6⇐ (e)
Remark
‘Spreadability’ implies ‘(full) conditional independence’ is the hard
part!
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An ingredient for proving full cond. independence
Localization Preserving Mean Ergodic Theorem (K ’08)
Let (M, ψ) be a probability space and suppose {αN }N∈N0 is a
family of ψ-preserving completely positive linear maps of M
satisfying
1. MαN ⊂ MαN+1 for all N ∈ N0 ;
W
2. M = N∈N0 MαN .
Further let
n−1
(n)
MN :=
1X k
αN
n
and TN :=
k=0
Y
~ N
(N)
αlN Ml .
l=0 l
Then we have
sot- lim TN (x) = EMα0 (x)
N→∞
for any x ∈ M.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Intermediate discussion
• Noncommutative conditional independence emerges from
distributional symmetries
For further details see:
C. Köstler. A noncommutative extended de Finetti theorem. J.
Funct. Anal. 258, 1073-1120 (2010)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Intermediate discussion
• Noncommutative conditional independence emerges from
distributional symmetries
• Exchangeability is too weak to identify the structure of the
underlying noncommutative probability space
For further details see:
C. Köstler. A noncommutative extended de Finetti theorem. J.
Funct. Anal. 258, 1073-1120 (2010)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Intermediate discussion
• Noncommutative conditional independence emerges from
distributional symmetries
• Exchangeability is too weak to identify the structure of the
underlying noncommutative probability space
• All reverse implications in the noncommutative extended de
Finetti theorem fail due to deep structural reasons!
For further details see:
C. Köstler. A noncommutative extended de Finetti theorem. J.
Funct. Anal. 258, 1073-1120 (2010)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Intermediate discussion
• Noncommutative conditional independence emerges from
distributional symmetries
• Exchangeability is too weak to identify the structure of the
underlying noncommutative probability space
• All reverse implications in the noncommutative extended de
Finetti theorem fail due to deep structural reasons!
• This will become clear from braidability...
For further details see:
C. Köstler. A noncommutative extended de Finetti theorem. J.
Funct. Anal. 258, 1073-1120 (2010)
R. Gohm & C. Köstler. Noncommutative independence from the
braid group B∞ . Commun. Math. Phys. 289, 435–482 (2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Artin braid groups Bn
Algebraic Definition (Artin 1925)
Bn is presented by n − 1 generators σ1 , . . . , σn−1 satisfying
σi σj σi = σj σi σj
if | i − j | = 1
(B1)
σi σj = σj σi
if | i − j | > 1
(B2)
0 1
ppp
i-1 i
ppp
0 1
ppp
i-1 i
ppp
Figure: Artin generators σi (left) and σi−1 (right)
B1 ⊂ B2 ⊂ B3 ⊂ . . . ⊂ B∞ (inductive limit)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Braidability
Definition (Gohm & K. ’08)
A sequence (xn )n≥0 in (A, ϕ) is braidable if there exists a
representation ρ : B∞ → Aut(A, ϕ) satisfying:
xn = ρ(σn σn−1 · · · σ1 )x0
for all n ≥ 1;
x0 = ρ(σn )x0
if n ≥ 2.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Braidability
Definition (Gohm & K. ’08)
A sequence (xn )n≥0 in (A, ϕ) is braidable if there exists a
representation ρ : B∞ → Aut(A, ϕ) satisfying:
xn = ρ(σn σn−1 · · · σ1 )x0
for all n ≥ 1;
x0 = ρ(σn )x0
if n ≥ 2.
Braidability extends exchangeability
• If ρ(σn2 ) = id for all n, one has a representation of S∞ .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Braidability
Definition (Gohm & K. ’08)
A sequence (xn )n≥0 in (A, ϕ) is braidable if there exists a
representation ρ : B∞ → Aut(A, ϕ) satisfying:
xn = ρ(σn σn−1 · · · σ1 )x0
for all n ≥ 1;
x0 = ρ(σn )x0
if n ≥ 2.
Braidability extends exchangeability
• If ρ(σn2 ) = id for all n, one has a representation of S∞ .
• (xn )n≥0 is exchangeable ⇔
Claus Köstler
n (x )
n n≥0 is braidable and
ρ(σn2 ) = id for all n.
Symmetries and Independence in Noncommutative Probability
Braidability implies spreadability
Consider the conditions:
(a) (xn )n≥0 is exchangeable
(b) (xn )n≥0 is braidable
(c) (xn )n≥0 is spreadable
(d) (xn )n≥0 is stationary and Atail -independent
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Braidability implies spreadability
Consider the conditions:
(a) (xn )n≥0 is exchangeable
(b) (xn )n≥0 is braidable
(c) (xn )n≥0 is spreadable
(d) (xn )n≥0 is stationary and Atail -independent
Theorem (Gohm & K. ’08)
(a) ⇒ (b) ⇒ (c) ⇒ (d),
Claus Köstler
but (a) 6⇐ (b) 6⇐ (d)
Symmetries and Independence in Noncommutative Probability
Braidability implies spreadability
Consider the conditions:
(a) (xn )n≥0 is exchangeable
(b) (xn )n≥0 is braidable
(c) (xn )n≥0 is spreadable
(d) (xn )n≥0 is stationary and Atail -independent
Theorem (Gohm & K. ’08)
(a) ⇒ (b) ⇒ (c) ⇒ (d),
but (a) 6⇐ (b) 6⇐ (d)
Observation
Large and interesting class of spreadable sequences is obtained
from braidability.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Braidability implies spreadability
Consider the conditions:
(a) (xn )n≥0 is exchangeable
(b) (xn )n≥0 is braidable
(c) (xn )n≥0 is spreadable
(d) (xn )n≥0 is stationary and Atail -independent
Theorem (Gohm & K. ’08)
(a) ⇒ (b) ⇒ (c) ⇒ (d),
but (a) 6⇐ (b) 6⇐ (d)
Observation
Large and interesting class of spreadable sequences is obtained
from braidability.
Open Problem
Construct spreadable sequence which fails to be braidable.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Intermediate Discussion
There are many examples of braidability!
• subfactor inclusion with small Jones index
(’Jones-Temperley-Lieb algebras and Hecke algebras’)
• left regular representation of B∞
• vertex models in quantum statistical physics (‘Yang-Baxter
equations’)
• ...
• representations of the symmetric group S∞
For further details see:
R. Gohm & C. Köstler. Noncommutative independence from the
braid group B∞ . Commun. Math. Phys. 289, 435–482 (2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
independence ←→
Claus Köstler
symmetries of
tensor products
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
Claus Köstler
symmetries of
tensor products
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
symmetries of
tensor products
Subject of distributional symmetries and invariance principles
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
symmetries of
tensor products
Subject of distributional symmetries and invariance principles
Free Probability
Voiculescu
(∼ 1982)
free
independence
Claus Köstler
←→
symmetries of
free products
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
symmetries of
tensor products
Subject of distributional symmetries and invariance principles
Free Probability
Voiculescu
(∼ 1982)
free
independence
←→
symmetries of
free products
Foundation of free probability theory
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
symmetries of
tensor products
Subject of distributional symmetries and invariance principles
Free Probability
K. & S.
(2008)
quantum
←→
exchangeability
Voiculescu
(∼ 1982)
free
independence
←→
symmetries of
free products
Foundation of free probability theory
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Is there a free analogue of de Finetti’s theorem?
Classical Probability
De Finetti’s
Theorem
exchangeability
←→
independence ←→
symmetries of
tensor products
Subject of distributional symmetries and invariance principles
Free Probability
K. & S.
(2008)
quantum
←→
exchangeability
Voiculescu
(∼ 1982)
free
independence
←→
symmetries of
free products
Foundation of free probability theory
New direction of research in free probability
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Key idea (K. & Speicher ’08)
• Rewrite exchangeability (for a classical sequence X ) as a
co-symmetry using the Hopf C*-algebra C (Sk )
For details see:
C. Köstler & R. Speicher. A noncommutative de Finetti theorem:
Invariance under quantum permutations is equivalent to freeness
with amalgamation. Commun. Math. Phys. 291(2), 473–490
(2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Key idea (K. & Speicher ’08)
• Rewrite exchangeability (for a classical sequence X ) as a
co-symmetry using the Hopf C*-algebra C (Sk )
• Apply ‘quantization’:
For details see:
C. Köstler & R. Speicher. A noncommutative de Finetti theorem:
Invariance under quantum permutations is equivalent to freeness
with amalgamation. Commun. Math. Phys. 291(2), 473–490
(2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Key idea (K. & Speicher ’08)
• Rewrite exchangeability (for a classical sequence X ) as a
co-symmetry using the Hopf C*-algebra C (Sk )
• Apply ‘quantization’:
Replace Xk ’s by operators x1 , x2 , . . .
For details see:
C. Köstler & R. Speicher. A noncommutative de Finetti theorem:
Invariance under quantum permutations is equivalent to freeness
with amalgamation. Commun. Math. Phys. 291(2), 473–490
(2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Key idea (K. & Speicher ’08)
• Rewrite exchangeability (for a classical sequence X ) as a
co-symmetry using the Hopf C*-algebra C (Sk )
• Apply ‘quantization’:
Replace Xk ’s by operators x1 , x2 , . . .
Replace C (Sk ) by quantum permutation group As (k)
For details see:
C. Köstler & R. Speicher. A noncommutative de Finetti theorem:
Invariance under quantum permutations is equivalent to freeness
with amalgamation. Commun. Math. Phys. 291(2), 473–490
(2009)
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum Permutation Groups
Definition and Theorem (Wang 1998)
The quantum permutation group As (k) is the universal unital
C*-algebra generated by eij (i, j = 1, . . . k) subject to the relations
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum Permutation Groups
Definition and Theorem (Wang 1998)
The quantum permutation group As (k) is the universal unital
C*-algebra generated by eij (i, j = 1, . . . k) subject to the relations
• eij2 = eij = eij∗ for all i, j = 1, . . . , k
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum Permutation Groups
Definition and Theorem (Wang 1998)
The quantum permutation group As (k) is the universal unital
C*-algebra generated by eij (i, j = 1, . . . k) subject to the relations
• eij2 = eij = eij∗ for all i, j = 1, . . . , k


e11 · · · e1k


• each column and row of  ... . . . ...  is a partition of unity
ek1 · · · ekk
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum Permutation Groups
Definition and Theorem (Wang 1998)
The quantum permutation group As (k) is the universal unital
C*-algebra generated by eij (i, j = 1, . . . k) subject to the relations
• eij2 = eij = eij∗ for all i, j = 1, . . . , k


e11 · · · e1k


• each column and row of  ... . . . ...  is a partition of unity
ek1 · · · ekk
As (k) is a compact quantum group in the sense of Woronowicz,
in particular a Hopf C*-algebra
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum Permutation Groups
Definition and Theorem (Wang 1998)
The quantum permutation group As (k) is the universal unital
C*-algebra generated by eij (i, j = 1, . . . k) subject to the relations
• eij2 = eij = eij∗ for all i, j = 1, . . . , k


e11 · · · e1k


• each column and row of  ... . . . ...  is a partition of unity
ek1 · · · ekk
As (k) is a compact quantum group in the sense of Woronowicz,
in particular a Hopf C*-algebra
The abelianization of As (k) is C (Sk ), the continuous functions on
the symmetric group Sk
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum exchangeability
Definition (K. & Speicher 2008)
Consider a probability space (A, ϕ).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum exchangeability
Definition (K. & Speicher 2008)
Consider a probability space (A, ϕ). A sequence of operators
x1 , x2 , . . . ⊂ A is quantum exchangeable if its distribution is
invariant under the coaction of quantum permutations:
ϕ(xi1 · · · xin )1lAs (k) =
k
X
ei1 j1 · · · ein jn ϕ(xj1 · · · xjn )
j1 ,...,jn =1
for all k × k-matrices (eij )ij satisfying defining relations for As (k).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Quantum exchangeability
Definition (K. & Speicher 2008)
Consider a probability space (A, ϕ). A sequence of operators
x1 , x2 , . . . ⊂ A is quantum exchangeable if its distribution is
invariant under the coaction of quantum permutations:
ϕ(xi1 · · · xin )1lAs (k) =
k
X
ei1 j1 · · · ein jn ϕ(xj1 · · · xjn )
j1 ,...,jn =1
for all k × k-matrices (eij )ij satisfying defining relations for As (k).
Remark
=⇒ exchangeability
quantum exchangeability ⇐=
6
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A free analogue of de Finetti’s theorem
Theorem (K. & Speicher 2008)
The following are equivalent for an infinite sequence of random
variables x1 , x2 , . . . in (A, ϕ):
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A free analogue of de Finetti’s theorem
Theorem (K. & Speicher 2008)
The following are equivalent for an infinite sequence of random
variables x1 , x2 , . . . in (A, ϕ):
(a) the sequence is quantum exchangeable
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A free analogue of de Finetti’s theorem
Theorem (K. & Speicher 2008)
The following are equivalent for an infinite sequence of random
variables x1 , x2 , . . . in (A, ϕ):
(a) the sequence is quantum exchangeable
(b) the sequence is identically distributed and freely
independent with amalgamation over T
Here T denotes the tail von Neumann algebra
\
T =
vN(xk |k ≥ n)
n∈N
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A free analogue of de Finetti’s theorem
Theorem (K. & Speicher 2008)
The following are equivalent for an infinite sequence of random
variables x1 , x2 , . . . in (A, ϕ):
(a) the sequence is quantum exchangeable
(b) the sequence is identically distributed and freely
independent with amalgamation over T
(in the sense of Voiculescu 1985)
Here T denotes the tail von Neumann algebra
\
T =
vN(xk |k ≥ n)
n∈N
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A free analogue of de Finetti’s theorem
Theorem (K. & Speicher 2008)
The following are equivalent for an infinite sequence of random
variables x1 , x2 , . . . in (A, ϕ):
(a) the sequence is quantum exchangeable
(b) the sequence is identically distributed and freely
independent with amalgamation over T
(in the sense of Voiculescu 1985)
(c) the sequence canonically embeds into FN
T vN(x1 , T ), a von
Neumann algebraic amalgamated free product over T
Here T denotes the tail von Neumann algebra
\
T =
vN(xk |k ≥ n)
n∈N
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Observation: Proof of this result uses Choquet theory.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Observation: Proof of this result uses Choquet theory.
Open Problem (Free Coin Tosses)
Let p1 , p2 , p3 . . . be a quantum exchangeable
sequence of
T
projections. Identify its tail algebra T = n∈N vN(pk |k ≥ n).
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Observation: Proof of this result uses Choquet theory.
Open Problem (Free Coin Tosses)
Let p1 , p2 , p3 . . . be a quantum exchangeable
sequence of
T
projections. Identify its tail algebra T = n∈N vN(pk |k ≥ n).
Remarks
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Observation: Proof of this result uses Choquet theory.
Open Problem (Free Coin Tosses)
Let p1 , p2 , p3 . . . be a quantum exchangeable
sequence of
T
projections. Identify its tail algebra T = n∈N vN(pk |k ≥ n).
Remarks
2
• This sequence is realized on FN
T vN(C , T ), the von Neumann
algebraic infinite free product with amalgamation over T .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
An open problem on the tail algebra
Theorem (Størmer 1969)
Symmetric states of infinite tensor products of C*-algebras are
convex combinations of infinite tensor product states.
Observation: Proof of this result uses Choquet theory.
Open Problem (Free Coin Tosses)
Let p1 , p2 , p3 . . . be a quantum exchangeable
sequence of
T
projections. Identify its tail algebra T = n∈N vN(pk |k ≥ n).
Remarks
2
• This sequence is realized on FN
T vN(C , T ), the von Neumann
algebraic infinite free product with amalgamation over T .
• Equivalent formulation of problem: Identify all symmetric
states of FN C2 , the infinite free product of C2 .
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A new application of exchangeability: Thoma’s theorem
A function χ : S∞ → C is a character if it is constant on
conjugacy classes, positive definite and normalized at the unity.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
A new application of exchangeability: Thoma’s theorem
A function χ : S∞ → C is a character if it is constant on
conjugacy classes, positive definite and normalized at the unity.
Theorem (Thoma ’64, Kerov & Vershik ’81, Okounkov ’97,
Gohm & K. ’09)
An extremal character of the group S∞ is of the form
χ(σ) =
∞
Y
k=2
mk (σ)

∞
∞
X
X

.
bjk 
aik + (−1)k−1
i=1
j=1
Here mk (σ) is the number of k-cycles in the permutation σ and
∞
the two sequences (ai )∞
i=1 , (bj )j=1 satisfy
a1 ≥ a2 ≥ · · · ≥ 0,
b1 ≥ b2 ≥ · · · ≥ 0,
∞
X
i=1
Claus Köstler
ai +
∞
X
bj ≤ 1.
j=1
Symmetries and Independence in Noncommutative Probability
Summary
• Out of the noncommutative extended de Finetti theorem
emerges a very general notion of noncommutative conditional
independence which invites to develop several new lines of
research along the classical subject of distributional
symmetries and invariance principles.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Summary
• Out of the noncommutative extended de Finetti theorem
emerges a very general notion of noncommutative conditional
independence which invites to develop several new lines of
research along the classical subject of distributional
symmetries and invariance principles.
• Braidability as a new symmetry extends exchangeability and
provides a new perspective on subfactor theory and free
probability theory
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Summary
• Out of the noncommutative extended de Finetti theorem
emerges a very general notion of noncommutative conditional
independence which invites to develop several new lines of
research along the classical subject of distributional
symmetries and invariance principles.
• Braidability as a new symmetry extends exchangeability and
provides a new perspective on subfactor theory and free
probability theory
• Quantum exchangeability gives a beautiful characterization of
freeness with amalgamation and opens a new research
direction in free probability. The free version of the de Finetti
theorem shows that a quantum symmetry leads to a quantum
probabilistic invariant.
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Summary
• Out of the noncommutative extended de Finetti theorem
•
•
•
•
emerges a very general notion of noncommutative conditional
independence which invites to develop several new lines of
research along the classical subject of distributional
symmetries and invariance principles.
Braidability as a new symmetry extends exchangeability and
provides a new perspective on subfactor theory and free
probability theory
Quantum exchangeability gives a beautiful characterization of
freeness with amalgamation and opens a new research
direction in free probability. The free version of the de Finetti
theorem shows that a quantum symmetry leads to a quantum
probabilistic invariant.
A new approach is opened to the theory of asymptotic
representations of symmetric groups.
...
Claus Köstler
Symmetries and Independence in Noncommutative Probability
Thank you for your attention!
Claus Köstler
Symmetries and Independence in Noncommutative Probability
References
C. Köstler. A noncommutative extended de Finetti theorem. J. Funct. Anal.
258, 1073-1120 (2010)
(electronic: arXiv:0806.3621v1)
R. Gohm & C. Köstler. Noncommutative independence from the braid group
B∞ . Commun. Math. Phys. 289, 435–482 (2009)
(electronic: arXiv:0806.3691v2)
C. Köstler & R. Speicher. A noncommutative de Finetti theorem: Invariance
under quantum permutations is equivalent to freeness with amalgamation.
Commun. Math. Phys. 291(2), 473–490 (2009)
(electronic: arXiv:0807.0677v1)
C. Köstler. On Lehner’s ‘free’ noncommutative analogue of de Finetti’s
theorem. 11 pages. To appear in Proc. Amer. Math. Soc.
(electronic: arXiv:0806.3632v1)
R. Gohm & C. Köstler. Noncommutative independence from characters of the
symmetric group S∞ . 40 pages. Preprint (2010).
Claus Köstler
Symmetries and Independence in Noncommutative Probability